SET 4 1. In a group of people, some are in favor of a tax increase on rich people to reduce the federal deficit and others are against it. Three persons are selected at random from this group and their opinions in favor or against raising such taxes are noted. How many total outcomes are possible? Write these outcomes in a sample space S. Draw a tree diagram for this experiment. 2. A Committee of persons engaged in real estate included 12 brokers . Of the 12 brokers, 3 sold both residential and commercial property, and 5 sold commercial property only. Draw a Venn diagram that shows how many brokers sold: a. Residential. b. Commercial. c. Residential only. 3. The characteristics of a sample of students at a midwestern university are as follows: Males Females Male undergraduates Male graduates Female undergraduate Female graduates 10 0 90 65 35 65 25 a. Set up a contingency table to show the relationships. b. What are the marginal probabilities? c. What are the joint probabilities? d. What are the conditional probabilities? e. Are sex and student status independent? 4. An office has 13 men and 7 women. Of these, 10 men and 4 women drive to work and the rest ride the bus to work. a. What is the probability that a person selected at random from the office rides the bus? b. What is the probability that a woman selected at random from the office rides the bus? 5. A survey of shoppers at a convenience grocery store showed that 70% purchase milk and 40% purchase bread. If 25% of all the shoppers purchase both milk and bread, a. What is the probability that a randomly selected shopper will purchase at least one of those items? b. What is the probability that a customer who has purchased bread will also purchase milk? c. What is the probability that a customer who has purchased milk will also purchase bread? d. Are the events "purchase milk" and "purchase bread" independent? Explain. 6. Given the employment data for the Cork Manufacturing Company below Locatio n Plant Office Sales TOTAL Mal e 1600 20 40 1660 Femal e 200 140 0 340 Total 1800 160 40 2000 Assume an employee is selected at random, and use the rules specified to find the probabilities requested: (a) Use the complement rule to find the probability of selecting a male. (b) Use the multiplication rule to find the probability of selecting a female who is an office worker. (c) Use the addition rule to find the probability of selecting either a female or a plant worker (or one who is both). (d) Are the categories employment and sex statistically independent? 7. The probability that an accountant will find an error in the Adams Corporation's accounts in a particular month is .05, and the probability that he will find an error in the Jones Corporation's accounts in the same month is .10. The two events are independent. (a) What is the probability that the accountant will find an error in Adam's accounts, but not in Jones's accounts in this month? (b) What is the probability that the accountant will find an error in the accounts of either Adams or Jones, or possibly both, during this month? (c) What is the probability that the accountant will find an error in either Adam's accounts or Jones's accounts, but not both, during this month? (d) What is the probability that the accountant will find an error in neither Adams's nor Jones's accounts during this month? 8. A company has two vacancies at the junior executive level. Ten people, seven men and three women, are eligible and equally qualified. The company has decided to draw two names at random from the list of eligible. What is the probability that (a) both positions will be filled by women? (b) at least one of the positions will be filled by women? (c) neither of the positions will be filled by a woman? 9. A market analyst assigns the following subjective probabilities for the Dow Jones Industrial averages at the close of tomorrow's market. Dow Jones Compared to Today Probability Higher (E1) 0.6 The Same (E2) 0.2 Lower (E3) 0.1 (a) Are the above probability assignments valid? (b) Assume that in the previous example an analyst predicts that the chance of the market going up tomorrow is 10 times as large as the chance of the market going down. Furthermore she feels that the chance of the market remaining the same is 9 times as large as the market going down. Assign valid probabilities to the markets outcome. 10. A firm interviews 15 applicants for a position on its sales force. One question on the application concerns leisure time activities. Six applicants say they spend a major portion of their leisure time playing golf. Ten mention bowling, three do not mention sports at all. How many applicants spend their leisure time both golfing and bowling? 11. A survey of automobile ownership was conducted for 200 families in Houston. The results of the study showing ownership of automobiles of U.S. and foreign manufacturers are summarized below: Yes Do you own a car Foreign car Yes 30 N o 10 No 150 10 Total 160 40 (a) Show the joint probability table for the above data? (b) Use the marginal probabilities to compare U.S. and foreign car ownership. (c) What is the probability that a family will own both a U.S. and a foreign car. (d) What is the probability that a family owns a car, U.S. or foreign? (e) If a family owns a U.S. car, what is the probability that it also owns a foreign car? (f) If a family owns a foreign car, what is the probability that it also owns a U.S. car? (g) Are U.S. and foreign car ownerships independent events? Explain. 12. The Adams Corporation receives telephone calls from its three salesmen, A, B, C. Salesman A makes 40 percent of the calls, whereas salesman B and C each make 30 percent of the calls . If salesman A calls there is a 10 percent chance he will ask the company's sales manager for assistance, whereas if salesman B or c call, there is only a 5 percent chance that either of them will ask for assistance. One of the three salesmen telephones and asks the sales manager for assistance, but the sales manager is from his office, and no one can remember which salesman called. a. What is the probability that salesman A called? b. What is the probability that it was not salesman C who called. 13. The Champion Chocolate Company marketing manager feels there is a .70 probability that marketing ten-pound boxes of chocolate rather than five-pound boxes will result in a sales increase (S) (rather than a decrease, S). A market research firm has agreed to test her strategy in selected cities, and inform the company if the test results are favorable (F) or unfavorable (F). Both firms recognize that such tests are not always accurate and that the results may be misleading. Based on previous market tests, the research company says there is a .60 probability of getting Favorable (F) results given the Sales (S) would increase [i.e., P(F/S) = .60]. Also, the probability of getting Favorable results followed by a Sales decrease is .20 [i.e.,P(F/S) = .20]. A test is made, and the results are Favorable. Given the test information, revise the probability of a sales increase. 14. A race driver uses Make A cars 50% of the time, Make B cars 30% of the time, and Make C cars 20% of the time. Of 25 races he has entered with Make A cars, he has won 5. In 15 races with Make B cars, he has won 4. In 10 races with Make C cars he has won 4. He has just won a race. What is the probability that the car he was driving was (1) Make A? (2) Make B? (3) Make C